It's true that waterjet cutting doesn't put much stress on the sides of the material that it's cutting compared to, say, milling. But vibrations from the cutting process can cause problems, especially near thin features. We created a couple sample pieces from aluminum 6061 to show just how thin you can make waterjet cut features. The photo above shows on the left a piece made from 1/4 inch (6.35 mm) thick material, and on the right you can see the same design cut from 1/16 inch (1.6 mm) stock.
You'll notice that the thinner material has one fewer bar. That's because the beefier material can hold thin features better than its skinnier counterpart. Thinner material is weaker for a given area and so it's more likely to vibrate when cut. You can see in the closeup below where the bar broke off. This bar was drawn to be 0.018 inches (0.46 mm) thick.
Even the thicker material had some trouble with a bar this thin. You can see that on the top face (the side where the waterjet stream first enters the material), the bar is missing some of its thickness and is just barely attached.
Below is a CAD drawing with the thinnest 3 bars. They are 0.0566, 0.0372 and 0.018 inches thick, respectively.
Keep in mind that these bars are only suspended from one side to the main body of the part. Connecting them to a bigger piece on both ends would help keep them stable when cutting. Also, different materials will behave differently when waterjet cut.
To summarize: narrow areas in your design under 0.018 inches (0.046 mm) will probably not work out. Keep feature thickness to 0.037 inches (0.95 mm) and above. If absolutely must have thin features, use thicker material.